1. Field of the Invention
The present invention relates to a data processing system having a floating point arithmetic unit and, more particularly, to a method and apparatus for performing floating point division and square root operations.
2. Description of the Related Art
Floating point arithmetic units have long be able to perform division and square root operations. Although dedicated division and square root circuitry has been used in data processing systems to perform division and square root operations, the current trend is to perform these operations using circuitry associated with multiplication and addition. Examples of data processing systems using a multiplier and adder to perform division or square roots are contained in European patent application 89313402.3 (EPO publication no. 0377992 A2), European patent application 85106938.5 (EPO publication no. 0166999 A2), and European patent application 82111929.4 (EPO publication no. 0111587 A1).
By eliminating dedicated division and square root circuitry from the designs of floating point arithmetic units, designers of data processing systems can not only reduce costs but also conserve board or die space. In these data processing systems, division and square root operations are performed using the multiplier and the adder of the floating point arithmetic unit. This allows more design time, power and area to be focused on the much more frequently used adder and multiplier hardware. Since it has been estimated that approximately only one-tenth of the operations performed in a data processing system involve a division or square root operation, such designs are feasible. However, a nagging problem with this approach is that division and square root operations take considerably longer to compute than do addition and multiplication operations. This difference in computation time is due to the fact that addition and multiplication operations are computed directly, while division and square root operations are computed indirectly with an iterative procedure.
Iterative procedures for division can be grouped into different classes depending on their iterative operator. One class uses substraction as the iterative operator (e.g., nonrestoring division), and another class uses multiplication as the iterative operator. The iterative procedures which use multiplication as the iterative operator are preferred because they compute the result much faster. More specifically, for division, these iterative procedures are used to obtain a reciprocal of the divisor, and then to obtain the quotient, the resulting reciprocal is multiplied by the dividend. There are two well known iterative methods which use multiplication as the iterative operator to determine the reciprocal, namely series expansion and Newton-Raphson. Of these iterative methods, the Newton-Raphson method is most often used. A general discussion on iterative procedures for division may be found in Waser and Flynn, Introduction to Arithmetic For Digital Systems Designers, New York, 1982.
The Newton-Raphson method is a particularly attractive computational method for a high speed computer having a floating point multiplier and a floating point adder-subtractor. The Newton-Raphson method is useful for not only division, but also for square root. With each iteration, the Newton-Raphson method converges quadratically to its result.
Conventionally, the Newton-Raphson iteration is used to obtain an approximation to the inverse of the denominator in the case of division or the inverse square root in the case of square root. Thereafter, each subsequent Newton-Raphson iteration increases the precision of the inverse value. Once the desired precision is reached, the inverse value is multiplied by the numerator in the case of division or the input argument in the case of square root. Although the conventional approach works quite well when the results are computed to no more precision than that offered by the hardware associated with addition and multiplication, the conventional approach is unsatisfactory when higher precision results are needed. Specifically, it takes an inordinate amount of time to perform the final multiplication (final inverse value times the numerator) for higher precision results.
Thus, there is a definite need for an improved technique for obtaining high precision results of division and square root operations that operates at a considerably faster speed than the conventional approach.